Math 25 Activity 6: The Slope and Y-Intercept of a Line
The second part of the activities this week explores graphing lines.
Imagine you have a marble on a track. For our purposes, the track is practically 2-dimensional (just
upward track or downward track) and made of mostly lines except for small curves when changing
direction (no loops).
Draw a track that has parts where the marble is travelling up and parts where it is travelling down.
Also, make some of the inclines more steep than other parts, similarly draw the declines.
1. Which are the steepest inclines and declines on the track you drew?
2. Which are the least steep inclines and declines on the track you drew?
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Answer the same questions for this hand-drawn example of a track.
3. Which are the steepest inclines and declines on the track you drew?
4. Which are the least steep inclines and declines on the track you drew?
The slope of the line is a numerical value, usually indicated by the letter “m”, that indicates the
steepness of a line and whether the line looks like it is declining or inclining. Visually you
determined that there is a difference in steepness and a difference between incline and decline, but
let’s explore how to tell the difference based on the numerical value that is the slope.
Consider the following graphed line. We can make a right triangle by using the points indicated on
the graph (-3,7) and (6,4).
5. How many units is the height of the triangle (we call this the “rise” because it is vertical)?
6. How many units is the length of the triangle (we call this the “run” because it is horizontal)?
7. What numerical value do you get when you take the rise and divide it by the run?
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Now, consider a third point on the line (0,6). We repeat the process making a new right triangle.
8. How many units is the height of the new triangle (rise)?
9. How many units is the length of the new triangle (run)?
10. What numerical value do you get when you take the rise and divide it by the run?
11. How does this number compare with your answer to question 5?
The slope of the line is defined to be the constant number calculated by taking any two unique
points on a line and dividing the rise by the run.
We also want to note here that because when we look at the graph, if this line were a track, then
the marble would roll downhill. Consider moving your finger along the legs of the triangle in the
first example. You could have started at the point (-3,7) and moved down 3 units, then right 9 units.
The word “down” really relates to a negative value for the rise while the word “up” relates to a
positive rise. The word “right” relates to a positive value for the run while the word “left” relates to
a negative run. The slope you calculated in question 5, which is rise divided by run, should be
negative because you would divide -3 by 9 which reduces to the fraction -1/3.
12. Describe the directions if you traced the triangle starting at the point (6,4) and traveled to (0,6)
using the words up/down and left/right.
13. Do you still come up with a negative slope starting at a different point?
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14. Use the definition of slope to find the slope of the following line. Draw the right triangle you used
to help you.
The formula for slope is m 
y 2  y1
where you label your points how you want, your first point
x 2  x1
would be ( x1 , y1 ) and your second point would be ( x2 , y 2 ) . The subtraction is what calculates the
difference between the points for both the rise and the run. Notice how the change in y values is in
the numerator of the fraction and the change in x values is in the denominator. Mathematically, this
symbolizes rise divided by run. The formula also accounts for the slope being negative or positive. If
the number comes out negative, then you should see a decline. If the number comes out positive,
then you should see an incline.
15. Draw a line on the graph below that is more steep than the given line and draw a second line
that is less steep than the given line. Calculate all three slopes.
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16. What pattern do you notice when you compare the steepness of the line and the slope?
When you have positive slopes, the closer the slope is to zero, the less steep the line. The larger the
slope the more steep the line appears. When you have negative slopes, the closer the slope is to
zero, the less steep the line. The more negative a number, the steeper the slope of the declining
line.
The last thing we want to consider about graphing lines in this project is the difference between
lines with the same slope. Slope helps us imagine what the line might look like on a blank piece of
paper, but when graphing lines there is one more important factor that we need to consider.
17. The following lines have the same slope. What is unique about each line that can distinguish it
from the other two lines?
There are multiple correct answers to question 17, but
mathematicians have decided to use the point where a
line crosses the y-axis to be the second piece of
pertinent information to graph a line. We call it the yintercept and it is represented using the letter “b”.
Circle the three y-intercepts and determine their values.
Therefore, after exploring the slope and the y-intercept,
you have the tools to write the equation of a line using
the slope-intercept formula y=mx+b. You should also be
able to take an equation in that form and graph it.
Without writing your name, tear off this slip and answer the following before the end of class and
turn it into your instructor:
Describe the slope of a line and how to find the slope as a numerical value.
Instructor!
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